Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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98
Chao
ref
2
pix=,
=
p,(t).
An additional condition must not be set in the area
of fluid flow-out. Therefore, the movement domain
is
not known in
advance when there
is
a transition
to
Lagrange coordinates
(5,
t)
The deduction of a priori bounds for density in this case gets
essentially difficult. Investigation of a flow problem began in
L55]
on the base of the Burgers model system. S.Ya. Belov
[30]
has recently enhanced these results, however, a general system
hasn't been studied yet
(1.1
1.
4.
A
new formulation of a problem on gas movement caused by the
movement of a solid wall (piston problem) has been suggested in
the work
[55].
On the one hand a problem with the condition of
adhesion to the wall
is
valid. On the other hand, actually,
if
a
piston
is
pulled out of gas with a rather large velocity, the gas
takes off the piston. !'hen there
is
a lack of viscosity in gas
there exist some examples
of
precise solutions of such problems
L117].
Therefore the following set up of a problem on a piston for
system
(1.1)
is
more physically based.
Let
gas at an initial moment
occupy a semi-axis
x<O
and the right boundary move by the law
x
=
e(t). Gas may move
so
that an extreme right particle
is
situa-
ted on the wall z(t)
the wall
is
possible when there arises a free boundary x
=
y(t)
on which correlations hold
and that's why ~l~=~(~)
=
dz/dt.
A
take off
It
is
reasonable to consider that the piston takes off,
if
strain
u=
p
-u-
z:
reaches the minimum value, zero, for instance, cor-
responding to the strain on a free boundary.
If
the piston doesn't
take off, strain on the boundary
is
non-negative. Besides, the
condition of non-fluidity through a solid wall can be formulated
in the form of inequality on
a
sought function y(t):
y(t)
<-
z(t).
In a general form the boundary value problem
is
lowing way:
it
is
required to find the solution
the domain-w
<
x
<
y(t),
t
E
[0,
T]
with an
x
=
y(t)
on which conditions are satisfied
set up
in
the fol-
of system
(1.1)
in
known boundary
(9.3)
The last equality means that either condition
Y=Z
Or
0.
Q=
0
condition on a free boundary are satisfied. Boundary value condi-
tions
(9.3)
contain inequalities, and the problem becomes one-
-sided. After the transition to Lagrange coordinates the boundary
x
=
y(t) becomes known and then for system
(1.10)
there appears a
roblem with boundary conditions for
x
=
0
of the following
form
a
x
is
a Lagrange variable):
Correctness
of
Boundary
Problems
99
Since the equation for velocity
u
is
parabolic the following
problem can be taken for
a
modal one:
au
,a
dU
---
-
(a(u)
-
I,
(x.
t)
E
0.
=
(-m.
0)
x
(0.
T).
a?
a,
>
0,
..
.
.
.
"
.
.~
.._.
.
..
at
ax
ax
(9.4)
au
t
au
t
ax
1
x=o
0
ax
o
-
5
0,
I
U(X,
T)dTlxzo
SO,
(
-
Problems
of
the form (9.4) have not been studied yet, therefore,
in
addition to suggested
set
up for equations
of
viscous gas
it
is
advisable to investigate problems of the form (9.4) and the
similar
ones for rather general evolution equations. The theorem on roblem
solvability for the Burgers model system has been proved in
f551,
but a uniqueness problem remains unsolved.
5.
In the investigation
of
problems with free boundaries
arises
another complicated problem connected with the possible degenera-
tion of equations when density
p
turns into zero. The condition
on a free boundary (in Lagrange coordinates)
vp
-
p
=
0
and
continuity equation
&A
=-I
am
brings to the correlation
ud2)
ax
p2
at
aP
1
-
+-pp=o
at
-
v
which
is
a common differential equation for the density on
a
free
boundary.
If
at an initial moment the density
is
highly greater
than zero,
it
keeps this property for all
t>
0.
But
if
for
t
=
0
on free boundary
p=
0
on free boundary then
p
=
0
for
t
>0
too. There appears
a
degeneration in momentum and energy
equations, the character of degeneration
is
unknown.
6.
It
has been assumed earlier, that the
state function
p(p,
S)
is
given for
all
p
E
(0,
m).
For example, the Clapeyron function
p
=
H
pO.Van der
Waals
equation of
state
is
more precise
in which there are two positive parameters
a and b;
a
is
pro-
portional to gas molecule cohesive force,
b
is
proportional to
molecular volume.
This
equation
is
defined for
0
<
p
<
b-'
therefore, when proving evaluations for density one must not only
obtain boundedness
p
but an evaluation with
a
constant earlier
given
p(x,
t)
<
b-'
.
In the case of isothermal movement
8
=const
>
>
Othis
evaluation
is
deduced
in
the same way as in
83.
In
a
general case calculations become too cumbersome and the problem
remains unsolved.
p
=
Rp0/(1
-
pb)
-
ap2
100
Chapter
2
7.
There
is
another problem which
is
of great interest. The results
concerning the existence of periodic, almost periodic and bounded
solutions on the whole time axis have been obtained in L140-143J.
In particular,
it
has been demonstrated that periodic solutions
don't exist in momentum equation of the system (1.1) with the right
side. However,
it
is
possible that periodic solutions can occur
if
additional heat souces are available i.e.
if
a periodic right side
is
introduced in energy equation.
8.
Besides, approximate methods of equation system solutions (1.1
1,
for example differential methods,
is
an important topic for dis-
cussion.
At
present time there are a lot of papers which suggest
different methods of solutions
of
the Navier
-
Stokes equations for
compressible gas (see
[Ill, 117, 137, 731 and references) but their
stability and convergence have not been proved.
9. Note an extremely interestixy and complicated problem of finite
transition for viscosity tending to zero. This problem attracts
attention of many mathematicians. The results obtained are parti-
cular cases: Hopf substentiated a finite transition in the Burgers
equation for incompressible liquids, then generalizations of wider
class of quasi-linear equations of the
first
order and also of
systems of ordinary equations, describing self-similar flows of
progressive wave type were given (see L117] and references).
A
finite transition for the viscous gas equation system has been
proved only for model cases, when the solution
is
smooth in equa-
tions without viscosity
[55,
1071. Papers [107, 141
red the problem on the approximation of equations
(
1
.lo)
of a
P
1.10)
E-
regulation
is
used
have conside-
arabolic system.
It
has been proved that
if
in
the
first
equation
ap
2F=Ep2a_(p-),
aP
-+p
at
-
ax ax ax
there exists for any
E
>O
a global solution of an approximating
system and for
E
+
0
sequence of solutions of regularized problem
converges to the solution of a finite problem.
10.
Finally, point out that problems when coefficients of viscosity
and thermal conductivity depend upon thermodynamic quantities have
not been sufficiently studied. Besides, of great interest
is
the
study of multi-component and multi- hase media models, and the
equations of magnetic gas dynamics ?see
[62,
94, 119, 159-1631].
101
CHAFTER
3
INITIAL-B9UNDARY VAL'JE PROBLEilS FOR THE NAVIZR-STOKES E31JATIONS
OF
A
MoriHoxot;zmous
~IISCOUS
INCOI~IPRESSIBLE
FLUID
A
mathematical investigation of the Navier-Stokes model of
a
vis-
cous incompressible liquid
was
initiated in the classical works by
J.
Leray
[175,
1761.
Various aspects
of
the theory
of
the Navier-
Stokes equations
for
a homogenious liquid were presented in the
monograph by
O.A.
Ladyzhenskaya
L88J.
The results discussed
in
the
present book refer to the case of an non-homogenous liquid, the
interest to
its
model condiditioned by
its
importance for applied
fields
of
hydrodynamics, for instance, for oceanology and hydro-
logy (see
LIOO,
1951).
It
should be noted that account taken of
the inhomogeneity enriches the theoretical analysis with
a
number
of peculiarities related to
an
additional non-linearity
of
the
major
terms
of
the equations. Besides generalizing the known
theorems from
[88]
for
the case
of
a
non-homogenous liquid, the
formulations
of
some
other boundary problems are considered,
namely,
of
one-side boundary problems.
1.
THE
EXISTENCE
OF
GENERALIZED
SOLUTIONS
1.
Formulation
of
the
first
boundary problem
Let us
consider
a
svstem
of
eauations for
a
non-homogenious vis-
cous incompressible"
1
iqu id
A
a<
-#
-#
+
+
at
p[
-
+
(U
*
V)U]
=p
Au
-vp
+pf,
--t
2
+
(2
.v)p
=
0,
div
u
=
0.
at
Let the motion take place in a bounded domain
impermiable boundary
r
of the class
C2.
On
r
tion
-#
U(X,
t>
=
0,
x
E
I',
t
E
LO,
T].
(1.1)
51
c
R3
with a fixed
the adhesion condi-
(1.2)
is
fulfilled.
At
the moment
of
time
t
=
0
the initial conditions
+
are given.
Let
us assume that
quirements
uo(x) and
po(x)
satisfy the
re-
102
Chapter
3
+
div uo
=
0,
u0lr
=
0,
o
<
m
5
po(x>
5
GI
<
m,
x
E
Q.
Let us call the formulation (1.1)
-
(1.3) the
first
boundary
problem or, for simplicity, problem
1.
While studying
it,
as was
the case in the theory of a homogenious liquid
1881,
use
will
be
made of the two spaces of the given on
Q
divergence equals
zer8.
Namely,
J(Q)
and
J1(Q)
the norms
Lz(Q>
and
'YI:(Q)
of a set
of
the finite in
51
and in-
finitely differentiated solenoidal vestor-functions.
Definition
1.1.
A
pair
of
functions
(u,
p)
zed solution of problem
1,
if
vector-functions,whose
are closures in
is
called a generali-
and
if
the integral identities
T
+--f
+
+
++
-t+
I
{(P%
mt+(u
vmz,Q-
p(u,Q)
0
J1
(a
1
+
(Pf,
Q12,Qldt
+
I
-t
I
(P,
'Pt
+
(U
V)cP),,,dt
+
(Po,
'P(OI),,Q
=
0
(1.5)
0
-9
are fulfilled at any
Q(t>
E
c'(0,
T;
J1(Q>>
and
cp(t>
E
C'(0,
T;
The abo e de inition
is
equivalent
to
another one, where at all
E
10,
Tqinstead of (1.4), (1.5) the following equalities are
valid
:
Wi(51))
such that
O(T)
=
0,
q(T)
=
0
Initial-Boundary Value Problems
103
at arbitrary ?(t>
E
C1(O,
T;
J1(Q)),
+(t)
E
C1(O,
T;
W:(Q)).
To
prove the equivalence of the two definitions
of
tho generalized
solution, let us set in (1.4) and (1.5)
-D
-D
@(X,
t)
=
Y(xi
t),
CP(X,
t>
=
h(t)+(x, t),
where(H(t),
h(t))
E
C1(O,
T),
H(0)
=
H(T)
=
h(0)
=
h(T)
=
0.
Then, by the definition of a generalized derivative, we have that
the functions
(p3,
Y)2,s2
and
(p,
$)2,Q
on
[o,
in which case
-D
are absolutely continuous
d
+
-4
-D
+
ilt
-
(P3,
=
(PU,
Yt
+
(u
*v
>Y12,Q
-
+
(Pfl
Y)2,Q
9
(1.6)
-D-D
-D-D
-
P(U,
Y)
J'
(a
d
-
(P,
$)2,Q
=
(PI
JI,
+
(u
*v>+),,,.
dt
(1.7)
Hence, in
(1.4)
and
(1.5)
we have
for any smooth
H(t)
and
h(t).
If
we set
H(T)
=
h(T)
=
0
and
H(0)
=
h(0)
=
1
then we come to the equalities
Integrating
get the identities
(1.4*)
and
{1.5*).
Inversely,
if
in the latter
we set
t,
=
T
in which case Y(T)
=
0,
+(T)
=
0,
formulas (1.4) and
(1
.5).
Let us
now
formulate the basic result as
fa
as the oolvability
of
problem
1
in the class of generalized solutions
is
concerned.
Theorem
1.1.
Let
f(x, t)
E
L~,
,(%I,
uo
E
J(Q),
po
E
L~(Q),
0
<
rn
5
po(x)
5
hi
<
zed 0olution of problem
1.
The proof consists of three etages: (a) construction of approxi-
mate solutions, (b) proof
of
their compactness,
(3)
realization
of
a limiting transition.
(1.6)
and
(1.7)
with respect to
t
from
0
to
to,
we
then we get
+
-f
m.
Then there exists at least one generali-
104
Chapter
3
2.
Construction of approximate solutions
Let
us
take in the space J(Q)
tions of the spectral problem
'1
the basis
{J,
}
from the eigeeunc-
p
A
;'
-
Q1
=
AIG1,
div
G1
=
0,
G1l,
=
0,
1
=
1,
2,
...
orthonormalized in
L2(Q).
Let us seek for the approximate solutions
+N
u
(x,
t),
N
=
1,
2,
...,
in the form
of
the finite
sums
with the unknown coefficients c
...,
N,
while the functions
p
(x,
t) are sought from the problem
(t)
E
C1(O,
T),
1
=
I,
2, ...
NN1
For
simplicity, let us assume here that
po(x)
E
C1(Q>, (In the ge-
0
N
geral case, when
p
E
Lco(Q)
in
(1.9)
we set
p
I
t=o
=
pi(x>,
where
{pg(X>}
verging
to
Po(X)
It
simplifies, but insignificantly, the course of the subsequent
considerations.
In order to determine the coefficients cNl(t)
require that the i$ntity
(1.42
is
~atlsfied~for3~
and
pN
on
all the functions
9
of type
where R(t)
E
C'(U,
T),
H(1')
=
0.
following relations must hold:
is
a
sequence of the functions from C'(Q) con-
in the norms of the spaces
L
(52))
I
5
q
<
9
in (1.8) let us
m(X,
t)
=
H(t)(bJ(x),
j
=
1,
2,...,
N,
This
requirement means that the
which are a system of
N
differential equations for cNl(t):
N
dC~~l
=jNl
+
j
c
yi
CN1
=
fil,j=i,
...,
N.
c-
@Nil 'Ni
'PI1
1=1
1=1
dt
i,
1=1
(1.11)
Both the coefficients and the right-hand part of this system are
determined by the formulas
Initial- Boundary
Value
Problems
105
The initial data for equations
(1.11
sion of the initial velocity
Go
can be taken
from
the ezpan-
with respect to the basis
{$J}
-Po0
uo
=
c
cj+
j=i
(1.12)
Let us prove that problem (1.8)
-
(1.12)
is
solvable.
Lemma 1.1.
For
any
N
7,
2,...there exists a unique solution
(zN,
PN>
respect to
N
of the problem (1.8)
-
(1.12). Moreover such uniform with
a priori estimates are valid:
(1.13)
Proof
of
the Lemma.
Prom (1.91,
as from an equation of the
first
order with respect to
p
(x,
t)
we get the presentation
N
PNh
t)
=
PO(Y%,
x,
t>l,=,)
,
-
=
+N
u
(Y,
TI,
Y
I,&
=
x*
where
?(T,
x,
t)
is
the solution of the Cauchy problem
dy
d-c
It
results from this presentation that
m
5
pN(x,
t)
5
hi.
Then, by
way
of
multiplying system (1.10) by
c
(t)
and of swing
it
with
respect
to
j
from
1
to
N,
we obtain the equality
Nj
N”N
”N
-‘N
(P
Cut
+
(u
-
v
>u
Using equation (1.9)
we deduce from
it:
Applying the Cauchy inequality to the right-hand part
and allowing for the relations
rn
5
pN
5
iLi
second estimate (1.13).
On
the basis of the above estimates and
the Shauder principle of a fixed let us prove the solvabi-
lity
of
the problem
(1.8)
-
(1.12?S)int’
L+et c(0,
‘r>
be a space
of
the continuous on[Oc
w
=
(wl(t),
-..,
w,(t))
with
the
norm
we easily get the
vector-functions
106
Chapter
3
Let us take a limited closed convex set
K
in the space
C(0,
T)
++
K
=
(wl
IwI
5
C,,
wi(0)
=
ci,
i
=
1,
2,
...,
N
},
where
C,
is
the constant from (1.131, and ci are the coefficients
of the expansion
uo
from (1.12). Let
wo
=
(w:,
w:,
...,
wo)be
N
an
arbitrary element
of
the set
K=
Let us construct the vector
of the linear problem
+N+
N
vo
=
C
w:
Qi
and find the solution
p
N
N
i=i
Pt
+
(To
.
v
>P
=
0,
P(t=o
=
PO(X).
-t
Since v0
E
C(0,
T;
C'(Q))
due to the known properties of the
function
Gi
(see
[88]),
the solution;
this
case
F(x,
t)
E
C1(q),
(.1
=#
x
(0,
T).
,
we find the vector v1
=
C
wi
4'
exists,
is
unique, and in
Through given;' and
N
-P
p
from the system of ordi-
i=
i
nary differential equations
which
is
a linearization of the equations (1.11):
where the
following
notations are used
N.
cy-t
-t.
N+
+1
+
The Cauchy data for the system (l.ll*) are taken
from
(1.12):
(1.12*)
Problem (l.ll*), (1.12*),
if
it
is
unambiguously solvable, makes
it
possible
to find the vector
w
=
(wl,
w2,
...,
w,>as au inage
Y
11 1
Initial. Boundary
Value
Problems
107
of the element
3'
E
K
under the action of
a
certain operator
A
:K
+
C(0,
T).Solvability
of
the linear Cauchy problem (1.11
-
-
(1.12) results from the theory
of
ordinary differential equa-.
tions,
if
we make sure that the matrix of the coefficientsoli(t)
is
non-degenerate. The proof of this fact
is
carried out through
that by contradiction: let
at
t
=
to
E
LO,
T]
we have det Ilai(t,)ll=
=
0.
Then there exist the numbers
A,,
A2,
. .
.
,
AN
not
all
equal
to zero and such that
N.
Multiplying the j-th equdity in the given system by
A.
and
su-
ming up with respect to
j
we get the equality
J
N
+-+
+
N
+.
(p(t0N,
Y)~,~
=
o
for
Y
=
c
A.+J
j=1
J
-t
wherefrom we have
Y!
0
which contradicts the linear indepen-
dence
of
the basis vectors
{$'}.
Thus, we have constructed the transformationA
:
li
C(0,
T)
the
fixed points of which, together with the corresponding functions
p
give the solution to the problem (1.8)-(1.12).Let us demon-
strate that the set
K
is
mapped by the operator
A
onto itself,
i.e.
IW
I
5
GI.
tion from (1.10*)
by
wl(t)
sum
it
up with respect to
j
and
repeat the procedure of obtainfng estimate (1.13). Finally, let us
check up whether the mapping
A.
K
+
Kis
compact, for which purpose
let
us
get the estimates as far as the derivatives
dw!/dt
concerned. Let us multiply the j-th equation from (1.10*)
by
dwl/dt and
sun
it
up with respect
to
j
:
N
-+I
For this purpose let us multiply the j-th equa-
J
is
J
J
Therefrom we have the relation
-1
As
far as the basis functions
{$
>are smooth,
llsollc(Q)
5
C(ii),
then, applying the Young inequality and integrating with respect
to
t
we get the estimate